Similar Polygons & Scale Factor

What similar actually means

Same shape is a precise claim, not a vibe. Every pair of corresponding angles must be congruent, and every pair of corresponding sides must have the same ratio. If even one ratio breaks, the polygons are not similar.

The scale factor is that shared ratio. If a rectangle with sides $2$ and $1$ is similar to a rectangle with sides $6$ and $3$, the scale factor from small to large is $\dfrac{6}{2} = 3$ — and the other pair confirms it: $\dfrac{3}{1} = 3$.

Direction matters. The scale factor from the small polygon to the large one is $3$; going from large to small it is $\dfrac{1}{3}$. The two are always reciprocals.

In the figure, the two triangles are similar: matching angle marks show the corresponding angles are congruent, while the larger triangle is a scaled-up copy of the smaller.

Setting up the proportion

The similarity statement tells you which sides correspond — match the letters by position. In $ABCD \sim WXYZ$, side $\overline{AB}$ corresponds to $\overline{WX}$, and $\overline{BC}$ corresponds to $\overline{XY}$, because the letters sit in the same spots.

To find a missing side, write two corresponding ratios and set them equal: $\dfrac{AB}{WX} = \dfrac{BC}{XY}$. Keep the same figure on top in both fractions, then cross multiply and solve.

The classic trap: adding instead of multiplying

Similarity is about multiplying by one scale factor, not adding the same amount to every side. A $6 \times 4$ rectangle and an $8 \times 6$ rectangle have sides that each grew by $2$, but $\dfrac{8}{6} \neq \dfrac{6}{4}$, so they are not similar.

Whenever a problem asks whether two figures are similar, check the ratios of corresponding sides. All equal — similar. Any mismatch — not similar, no matter how alike the figures look.

Worked examples

Try one yourself

Common questions